Gears

A set of gears is installed on the plane. You are given the
center coordinate and radius of each gear, which are all
integer-valued. For a given source and target gear, indicate
what happens to the target gear if you attempt to turn the
source gear. Possibilities are:

The source gear cannot move, because it would drive some
gear in the arrangement to turn in both directions.

The source gear can move, but it is not connected to the
target gear.

The source gear turns the target gear, at a certain
ratio

If the source gear cannot move, give this result, even if
the source and target gears are not connected.

Input

Each input will consist of a single test case. Note that
your program may be run multiple times on different inputs. The
first line of input contains a single integer $n$ ($1
\le n \le 1000$), the total number of gears. Following
this will be $n$ lines,
one per gear, containing three integers: the $x, y$ ($-10\, 000 \le x, y \le 10\, 000$) and
$r$ ($1 \le r \le 10\, 000$) values for the
gear, where $(x,y)$ is the
position of the axle of the gear, and $r$ is its radius. Assume that the
teeth of the gears are properly designed, and accounted for in
the radius, so that any gear will mesh with any other gear if
(and only if) they are tangent to each other. The gears will
never overlap. The source gear is the first gear, the target
gear is the last gear specified.

Output

Output a single line, with the following content, based on
the result:

-1 if the source gear cannot
move.

0 if the source gear can move
but is not connected to the target.

a b if the source gear moves
the target gear, where $a$ and $b$ are two space-separated
integers, and $a:b$ is
the ratio of source gear revolutions to target gear
revolutions reduced to its lowest form (i.e. they have no
common factor other than $1$).

$a$ is always
positive.

If the target turns in the same direction as the
source, $b$ is
positive.

If the target turns in the opposite direction as the
source, $b$ is
negative.